Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

learn more… | top users | synonyms

3
votes
1answer
303 views

Prove $\sum_{n= -\infty}^{\infty} \frac{1}{(t+n)^2} = \frac{\pi^2 }{\sin^2(\pi t)}$ using f(x)=1-|x| and Poisson summation formula

I'd like to prove $$\sum_{n= -\infty}^{\infty} \frac{1}{(t+n)^2} = \frac{\pi^2 }{\sin^2(\pi t)}$$ by using the Poisson summation formula. There is a way to do it by firstly taking the Fourier ...
0
votes
1answer
214 views

Convolution of a sum of shifted delta functions

In the lecture notes for Fourier Transforms and it's Applications on page 212 by Bracewell he talks about representing a signal as a sum of distributions evenly spaced out by a distance p. $$\rho_p(x)...
0
votes
1answer
46 views

Representing a real sampled signal with N samples as a complex sampled signal with N/2 samples

I am studying the discrete Fourier transform, and in its most basic definition it is an invertible linear transformation on the complex numbers. From Wikipedia: The sequence of $N$ complex numbers $...
3
votes
1answer
50 views

Can't extract odd function with FFT

I can't correctly extract spectrum from data points of odd function (e.g. $\cos\left(\frac23\pi x\right)$, $16$-points vector $[1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1]$), instead of one function I get a ...
2
votes
0answers
71 views

Formula for trace of particular operators

Let $\mathcal{H}$ be the Hilbert space $L^2(\mathbb{R})$. View the Fourier transform as a unitary operator $\mathcal{F} \in B(\mathcal{H})$. For each function $f \in C_0(\mathbb{R})$, let $T(f) \in B(...
0
votes
1answer
194 views

Meaning of multiplication by $\sin$ in $\omega$-domain

Multiplying some signal, a function of time, $m(t)$ by a cosine $\cos{\omega' t}$ causes a shift in frequency of $m(t)$, by $\pm\omega'$. But what about multiplication by a sine wave, such $m(t)\cdot\...
0
votes
1answer
24 views

Showing that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set

I have the following problem: Show that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set in $L^2(D)$, where $D$ is any square whose sides have length ...
0
votes
1answer
73 views

Inverse Fourier Transform

I need help solving the following Fourier transform question. Given, $$ X_s(f) = \frac{1}{\Delta T} \sum_{n = -\infty}^{\infty} X\left(f - \frac{n}{\Delta T} \right) $$ $$ H(f) = \begin{cases} 1 &...
0
votes
0answers
111 views

Symbol of self-adjoint pseudodifferential operator

It seems that the following result should hold, but I can't find it explicitly anywhere. If $A=A^*$ is a properly supported pseudodifferential operator, does this imply that $\sigma_A(x,\xi)=\sigma_{A^...
1
vote
1answer
30 views

Fourier transform of $|u|^2*u$

Given the Fourier transform of $u$ is $\widehat{u}$, then what can I say about the Fourier transform of $|u|^2u$? Can I represent it by $\widehat{u}$?
1
vote
1answer
42 views

intuition behind inverse transform of $ cos(\omega_{0} t)$

Hi: After looking around the internet and looking at solutions to similar questions, I was finally able to convince myself of the following mathematically. If $G(\omega_{0}) = cos(\omega_{0} t)$, ...
2
votes
1answer
36 views

Problem about average of cos square (nt) where n is arbitrary

I often see people just say time average of cos^2(nwt) is 1/2, I want to know in what cases this is not valid? w is just the frequency, can be assumed as a constant. Assuming you are always ...
1
vote
0answers
180 views

Easiest way to prove integral of $e^{ikx}$ is $\delta(k)$

What is the easiest way to to derive the following equation: $$\int_{-\infty}^{\infty}e^{ikx}dx = 2\pi\delta(k)$$ I understand the equation can be derived by assuming the Fourier integral theorem, ...
1
vote
2answers
156 views

trigonometric interpolation of a sampled signal

Given N sampled points, using the FFT we can get the Fourier transform of those N points $X_k$. With N/2 the Nyquist frequency and $X_0$ the DC value. Using the inverse we can then get back the ...
1
vote
1answer
163 views

Characteristic Function Inversion

I am studying the relationship / bijection between characteristic functions and CDFs. In particular, given a characteristic function $\phi$ it is posible to recover the cumulative density function ...
1
vote
1answer
33 views

Can one express $f'(x)$ with the same basis as one uses for $f$?

If I have an orthonormal basis $\{\phi_n\}_1^\infty$ in space $L^2(a,b)$ and the generalized Fourier series expansion for $f$ would be: $$f= \sum \langle f, \phi_n\rangle\phi_n,$$ then can one use ...
0
votes
1answer
51 views

Show that $f(x)$ is orthogonal to $f'(x)$ in $L^2(-\pi, \pi)$

I have the following problem: Suppose $f$ is of class $C^{(1)}$, $\;2\pi$-periodic, and real-valued. Show that $f'$is orthogonal to $f$ in $L^2(-\pi, \pi)$ by a) expanding $f$ in (...
0
votes
1answer
520 views

Convolution of L1 & L2 function: definition

A book that I'm reading makes the following statement that I'm not sure how to understand: On $\mathbb R^n$, if $f\in L1$ and $g\in L2$, we have: $$\widehat{f*g}=\hat f \hat g$$ How do I read it? I ...
2
votes
1answer
41 views

Proving that $\langle f, g\rangle = \sum_n \langle f, \phi_n \rangle \overline{\langle g, \phi_n \rangle}$

I have the following problem to solve: If the set of functions $\{\phi_n \}_1^\infty$ is an orthonormal basis in $L^2(a,b)$ and the functions $f, g \in L^2(a,b)$, then show that: $$\...
3
votes
1answer
77 views

What are the steps to derive the following inverse Fourier transformations

I'm reading a text which is an introductory text on Fourier transforms. The author has two expressions: $$ F(\omega_{o}) = \frac{1}{\sigma \sqrt {2 \pi} } e^{\Large- \frac{{\omega_{o}}^2}{2\sigma^2}}...
1
vote
3answers
149 views

Understanding dot product of continuous functions

I'm reading about Fourier analysis and in my book the author speaks about dot product for continuous functions $f, g\in L^2(a,b)$(the set of functions which are square-integrable on the interval $[a,b]...
5
votes
0answers
48 views

If $f$ is $2 \pi$ periodic and $\int_{0}^{2 \pi} f(t) dt=0$ then $\int_{0}^{2 \pi} (f(t))^2 dt \le \int_{0}^{2 \pi} (f'(t))^2 dt$ [duplicate]

Given $f$ a real differentiable function, $2 \pi$ periodic such that $\int_{0}^{2 \pi} f(t) dt=0$ show that: $\int_{0}^{2 \pi} (f(t))^2 dt \le \int_{0}^{2 \pi} (f'(t))^2 dt$. When does equality hold?
2
votes
0answers
44 views

Finding the sum of a trigonometric series, fourier series

I need to compute that for $x \in [0, 2\pi]$ $$\sum_{n=1}^\infty\frac{\sin(nx)}{n^3} = \frac{1}{12}x(x-\pi)(x-2\pi)$$ by using the uniform convergence $$\sum_{n=1}^\infty\frac{\sin(nx)}{n} = \frac{\...
2
votes
0answers
27 views

A proof regarding Fourier-Polynoms

I want to prove the following: Let $f:\mathbb{R}\rightarrow \mathbb{C}$ so that $f \big |_{[0,2\pi]}$ is integrable. Let $V$ be the vectorspace of all $2\pi$-periodic functions and $U \subset V$ be ...
1
vote
3answers
1k views

Calculating own dft via matlab?

We are asked to code our own dft function from the formula : If everything is done correctly it should give the same result with matlab's own dft function, in the end I'm comparing them but they ...
2
votes
1answer
134 views

a generalization of normal distribution to the complex case: complex integral over the real line

How to prove $\int_{\mathbb{R}} e^{-\frac{(x+it)^2}{2}}dx=\sqrt{2\pi}$ for any $t\in \mathbb{R}$? I only obtained the case that $t=0$, $\int_{\mathbb{R}} e^{-\frac{x^2}{2}}dx=\sqrt{2\pi}$. Thanks.
5
votes
1answer
436 views

When is a Fourier series analytic?

By Fourier theory, every continuously differentiable function $f : S^1 \to \mathbf C$ admits a unique, uniformly convergent Fourier expansion $$f(\theta) = \sum_{n\in \mathbf Z} a_n e^{in\theta}.$$ ...
1
vote
2answers
52 views

Show that there does not exist $f\in L^2(R)$ such that $\overline{{\rm span}\{f(\cdot-n):n\in Z\}}=L^2(R)$

Show that there does not exist $f\in L^2(R)$ such that $$\overline{{\rm span}\{f(\cdot-n):n\in Z\}}=L^2(R).$$ In other words, for any square function $f$, the space of the span of all shifts of $f$...
0
votes
1answer
113 views

Integral is equal to $0$

Let be $f \in L^1[0,1]$, then it applies $ \int_0^1 \exp(2i\pi xk)f(x n)\,dx=0$ for $n,k\in \mathbb{N}$ with $0<k<n$. Ideas: f can be extended to a function on $\mathbb{R}$ with period $1$, ...
3
votes
1answer
147 views

What is the theory behind Fourier transform of “bad” (e.g. unbounded) functions?

When I was first introduced to Fourier transform, its core was a formula for it, something like: $$\tilde f(k)=\int_{-\infty}^{\infty} e^{-2\pi i kx}f(x)\text{d}x.\tag1$$ It works nice for good ...
2
votes
0answers
52 views

Can a continuous function on a compact group $G$ be interpreted as the sum in $C(G)$ of its Fourier series?

For a given function $f\in C(G)$ on a compact group $G$ its Fourier transform is defined as the family of operators $$ \widehat{f}_\sigma=\int_Gf(t)\cdot\sigma(t^{-1}) \ \text{d}\ t,\quad \widehat{f}_\...
3
votes
1answer
113 views

Solve $\mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ] $ using contour integration

I wish to evaluate $y(t) = \mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ] $, where $\mathscr{F}$ represents the Fourier transform, and U(t) represents the ...
0
votes
1answer
32 views

what is the mathematical reason for slow functions having high spectral density at low frequencies and vice versa

Hi : I'm reading an introductory book on Fourier transforms. After explaining the forward and inverse transformation clearly, the author then states: " We realize the dual character of the forward ...
1
vote
1answer
35 views

How do I deal with a seemingly fractional delays in discrete time fourier transforms?

Is a transfer function of a discrete time system is $H(e^{j\Omega})=e^{-j\Omega/4}$ and I feed it an impulse, what will be it's response? I know that technically a transfer function of $e^{j{\Omega}...
1
vote
1answer
56 views

Exists $C$ constant: $||f||_{L^{\infty}(\mathbb{R})}\leq\{ ||f||_{L^2(\mathbb{R}}+||f'||_{L^2(\mathbb{R})} \}, \forall f\in S(\mathbb{R})$

Show that there exists $C$ constatant such that $||f||_{L^{\infty}(\mathbb{R})}\leq\{ ||f||_{L^2(\mathbb{R}}+||f'||_{L^2(\mathbb{R})} \}, \forall f\in S(\mathbb{R})$. This is a question in my ...
2
votes
2answers
207 views

Greens function of 1-d forced wave equation

[ORIGINAL PROBLEM] You are given hat the Green's function $g(x,t,\xi, \phi)$ is $\frac{\partial^2g}{\partial t^2} - \frac{\partial^2g}{\partial x^2}=\delta(t-\tau)\delta(x-\xi)$ with $g(x,0,\xi, \...
0
votes
1answer
283 views

Fast convolution with striding

I want to convolve two discrete functions $f$ and $g$ using convolution stride size $a$ to get the result as $s_{a, i}$: $$s_{i,a} = \sum_i g_k f_{ai-k}$$ I know that simple convolution with $a=1$ ...
4
votes
1answer
111 views

Finding the period of the solution to $y'(x) = y(x) \cdot cos(x + y(x))$ with Fourier transform; how to interpret complex result?

A question elsewhere on this site asks about detecting the frequency of oscillations in a system defined by differential equations. The equation is $y'(x) = y(x) \cdot cos(x + y(x))$. The solution ...
1
vote
0answers
90 views

The Fourier sine transform of $f(x)/\sin x$

Is the following result $$\lim_{\lambda \to \infty} \frac{2}{\pi} \int_0^\infty \frac{f(x)}{\sin x} \sin(\lambda x) \, dx = f(0) + 2\sum_{k = 1}^\infty f(k\pi),$$ where $\lambda$ is an odd integer, ...
2
votes
1answer
61 views

the delta function written as the integral of a complex number

Hi: I've been reading an introductory book on Fourier transforms. The author explains the $\delta$ function ( while noting that it's really a distribution ) in the following manner which makes a lot ...
1
vote
1answer
72 views

Identically distributed and same characteristic function

If $X,Y$ are identically distributed random variables, then I know that their characteristic functions $\phi_X$ and $\phi_Y$ are the same. Does the converse also hold?
1
vote
0answers
159 views

Square-summable sequence and Fourier series

Every square-summable sequence $(a_{n})_{n}$ is represented by $a_{n}=\widehat{f}(4^n)$, where $\widehat{f}(i)$ is Fourier coefficient of continuous function $f$. Where can I find proof of this ...
1
vote
0answers
236 views

Autocorrelation Function and Power spectrum from ACF

In my assignment I am required to write or use a C code to find the autocorrelation function of a given function and then find the power spectrum from it. The function is as follows: $$f(t) = \cos(10 ...
1
vote
0answers
28 views

Vector*Matrix multiplication through Fast Transforms

I have recently read a paper in which the authors indicated they used a Fast Cosine Transform to implement a Vector*Matrix multiplication. The idea is to decrease complexity when implementing such ...
1
vote
0answers
150 views

Limit of an integration formula

Let $f$ be a smooth real (or complex) valued function defined on $S^2$. Then a direct calculation shows that $$\int_{S^2}f(x)e^{ixy}\, dx=-\frac{i}{\|y\|}e^{i\|y\|}f\left(\frac{y}{\|y\|}\right)+\frac{...
1
vote
2answers
399 views

Fourier transform of $e^{-|t|}\sin(t)$

How can i calculate the Fourier transform of $e^{-|t|}\sin(t)$. I guess I need to do something with convolution, but I am not sure. Can somebody show me the way?
4
votes
1answer
167 views

Let $f(x) = |\cos(x)|$. Prove the corresponding Fourier series converge point-wise or uniform and show identity.

Consider $f(x) = |\cos(x)|$ for $x \in \mathbb R$. I've proved the n'th fourier coefficient $c_n = \int^{\pi}_{-\pi} f(y)e^{-iny} \ dy = \frac 1 {2\pi} \frac {(-1)^{n-1}} {n^2-\frac 1 4}$. However, ...
1
vote
1answer
49 views

Question about the weak (1,1) bound for the Hilbert Transform (Javier Duoandikoetxea's Fourier Analysis)

I've been reading Duoandikoetxea's book, and, in chapter 3, he proves a weak (1,1) bound for the Hilbert transform. To set my question up, I'll outline the argument, and then point out where I'm ...
2
votes
1answer
274 views

$\sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90}$ [duplicate]

How we can do this sum? $$ \sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90} $$ I know that we could possibly use a Fourier series decomposition however I don't know what function to start with. I ...
1
vote
2answers
95 views

How to integrate this fourier transform?

I want to integrate $$\int_{-\infty}^{\infty} \frac{e^{itx}}{{1+x^2}} dx.$$ I don't see how substitution or integration by parts could help here. Does anybody know how to do this?